← Back 🏠 Home

Mathematics: Class 10 — Chapter 12: Surface Areas and Volumes

12.1 Introduction

In day-to-day life, we frequently encounter solids that are combinations of basic three-dimensional shapes such as cuboids, cones, cylinders, spheres, and hemispheres.

12.2 Surface Area of a Combination of Solids

To find the total surface area (TSA) of a combined solid, we sum the visible curved surface areas (CSA) of the individual constituent solids. Surface areas that are joined together and hidden inside the solid are excluded.

Key Formula Configurations
  • Capsule / Oil Tanker (Cylinder with two hemispherical ends):
    TSA = CSA of cylinder + 2 × CSA of hemisphere
    TSA = 2πrh + 2(2πr²) = 2πr(h + 2r)
  • Toy / Lattu (Cone surmounted on a hemisphere):
    TSA = CSA of cone + CSA of hemisphere
    TSA = πrl + 2πr² = πr(l + 2r)
    Note: Slant height of cone l = √(r² + h²)
  • Decorative Block (Cube surmounted by a hemisphere):
    TSA = TSA of cube - Base area of hemisphere + CSA of hemisphere
    TSA = 6a² - πr² + 2πr² = 6a² + πr²

Worked Examples
  • Example 1 (Toy): A playing top of total height 5 cm and diameter 3.5 cm (radius r = 1.75 cm) consists of a cone on a hemisphere. Height of cone h = 5 - 1.75 = 3.25 cm. Slant height l = √(1.75² + 3.25²) ≈ 3.7 cm. Total area to colour ≈ 39.6 cm².
  • Example 2 (Decorative Block): A cube of edge 5 cm is surmounted by a hemisphere of diameter 4.2 cm (r = 2.1 cm). Total surface area = 6(5)² + π(2.1)² = 150 + 13.86 = 163.86 cm².
12.3 Volume of a Combination of Solids

Unlike surface area, the volume of a combined solid is straightforward: it is simply the sum of the volumes of its constituent parts.

Key Volume Formulas
  • Volume of Combined Solid = Volume of Solid 1 + Volume of Solid 2
  • Cylinder: V = πr²h
  • Cone: V = (1/3)πr²h
  • Sphere: V = (4/3)πr³
  • Hemisphere: V = (2/3)πr³

Worked Examples
  • Example 5 (Shed): A cuboidal shed (15 m × 7 m × 8 m) surmounted by a half-cylinder of diameter 7 m and length 15 m. Total volume = Volume of cuboid + 1/2 (Volume of cylinder) = (15 × 7 × 8) + 1/2 (π × 3.5² × 15) = 1128.75 m³.
  • Example 6 (Glass with Hemispherical Raised Bottom): A cylindrical glass of inner diameter 5 cm and height 10 cm has a hemispherical raised bottom. Apparent capacity = πr²h = 196.25 cm³. Actual capacity = Apparent capacity - Volume of hemisphere = 196.25 - (2/3)πr³ = 196.25 - 32.71 = 163.54 cm³.
12.4 Summary
  • To determine the surface area of a combined solid, identify the exposed surfaces of the individual shapes and sum their areas.
  • To find the volume of a combined solid, sum the individual volumes of the constituent shapes.